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Question Number 128998 by bramlexs22 last updated on 12/Jan/21

$$\mathrm{Given}\sin \mathrm{x}+\cos \mathrm{x}=\frac{5}{6}.\mathrm{Find}$$$$\mathrm{the}\mathrm{value}\mathrm{of}\frac{1}{\sin \mathrm{x}}+\frac{1}{\cos \mathrm{x}}.$$

Answered by liberty last updated on 12/Jan/21

$$\mathrm{From}\mathrm{condition}\sin \mathrm{x}+\cos \mathrm{x}=\frac{5}{6}$$$$\mathrm{we}\mathrm{get}1+2\sin \mathrm{x}\cos \mathrm{x}=\frac{25}{36}$$$$2\sin \mathrm{x}\cos \mathrm{x}=-\frac{11}{36}\mathrm{or}\sin \mathrm{x}\cos \mathrm{x}=-\frac{11}{72}$$$$\mathrm{so}\frac{1}{\sin \mathrm{x}}+\frac{1}{\cos \mathrm{x}}=\frac{\sin \mathrm{x}+\cos \mathrm{x}}{\sin \mathrm{x}\cos \mathrm{x}}=\frac{5}{6}\times (-\frac{72}{11})$$$$=-\frac{60}{11}$$

Answered by MJS_new last updated on 12/Jan/21

$$s+c=\frac{5}{6}$$$${(s+c)}^2=\frac{25}{36}\Rightarrow s^2+c^2+2sc=\frac{25}{36}\Rightarrow sc=-\frac{11}{72}$$$$\frac{1}{s}+\frac{1}{c}=\frac{s+c}{sc}=\frac{\frac{5}{6}}{-\frac{11}{72}}=-\frac{60}{11}$$

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